Congestion bounds via Laplacian eigenvalues and their application to tensor networks with arbitrary geometry

Embedding the vertices of arbitrary graphs into trees while minimizing some measure of overlap is an important problem with applications in computer science and physics. In this work, we consider the problem of bijectively embedding the vertices of an $n$-vertex graph $G$ into the \textit{leaves} of an $n$-leaf \textit{rooted binary tree} $\mathcal{T}$. The congestion of such an embedding is given by the largest size of the cut induced by the two components obtained by deleting any vertex of $\mathcal{T}$. We show that for any embedding, the congestion lies between $\lambda_2(G)\cdot 2n/9$ and $\lambda_n(G)\cdot n/4$, letting $0=\lambda_1(G)\le \cdots \le \lambda_n(G)$ be the Laplacian eigenvalues of $G$, and there is an embedding for which the congestion is at most $\lambda_n(G)\cdot 2n/9$. Beyond these general bounds, we determine the congestion exactly for hypercubes and lattice graphs, and obtain asymptotically tight bounds for random regular graphs and Erd\H{o}s-R\'{e}nyi graphs. We further introduce an efficient contraction procedure based on spectral ordering and dynamic programming, which produces low-congestion embeddings in practice. Numerical experiments on structured graphs, random graphs, and tensor network representations of quantum circuits validate our theoretical bounds and demonstrate the effectiveness of the proposed method. These results yield new spectral bounds on the memory and time complexity of exact tensor network contraction in terms of the underlying graph structure.